The spherical Paley-Wiener theorem on the complex Grassmann manifolds SU$(p+q)/$S$($U$_p\times$ U$_q)$

We prove the Paley-Wiener theorem for the spherical transform on the complex Grassmann manifolds $U/K=$SU$(p+q)/$S$($U$_p\times$   U$_q)$. This theorem characterizes the $K$-biinvariant smooth functions $f$ on the group $U$ that are supported in the $K$-invariant ball of radius $R$, with $R$ less than the injectivity radius of $U/K$, in terms of holomorphic extendability, exponential growth, and Weyl invariance properties of the spherical Fourier transforms $\hat{f}$, originally defined on the discrete set $\Lambda_{sph}$ of highest restricted spherical weights.